$\begingroup$"Now note that (2−3√)k<1 implying 1−(2−3√)k is fractional part of (2+3√)k " can u further explain this part$\endgroup$
– AmitDec 20 '13 at 16:12

$\begingroup$@Amit, together with the line before, it follows that $2a$ is the smallest integer greater than $(2+\sqrt3)^k$, and hence the greatest integer smaller than $(2+\sqrt3)^k$ is $2a-1$. So the fractional part is $(2+\sqrt3)^k-(2a-1)$.$\endgroup$
– Carsten SDec 20 '13 at 16:23